Higher order energy decompositions and the sum-product phenomenon.

Abstract: In 1983, Erdos and Szemeredi conjectured that either $|A+A|$ or $|AA|$ is at least $|A|^2$, up to a power loss. We make progress towards this conjecture by using various energy decomposition results, in a similar spirit to the recent Balog-Wooley decomposition. Our main tool is the Szemeredi-Trotter theorem from incidence geometry. For more information, see my blog which contains a video introduction the subject: gshakan.wordpress.com

Behavioral Contagion Type in Coupled Disease-Behavior Models

Matt Osborne (Mathematics, Ohio State University)

Abstract: Disease and behavior have long been recognized as coupled contagions, and mathematical models treating them as such have existed since the mid 2000s. In these models behavior and disease are often both classified as 'simple contagions'. However, the means of behavior spread may in fact be more complex. Using some tools of dynamical systems we investigate the difference between a coupled contagion model with behavior as a 'simple contagion' and one with behavior as a 'complex contagion'. We find that behavioral contagion type can have a significant impact on behavior-disease dynamics which can in turn have implications for potential public health interventions.

Compactness of the branch set for quasiregular mappings and mappings of finite distortion

Rami Luisto (Charles University, Prague)

Abstract: Quasiregular mappings and mappings of finite distortion are natural generalizations of holomorphic mappings to higher dimensions. Whereas the pointwise derivatives of holomorphic mappings map circles to circles, QR-maps and MFD are defined by requiring that the differential maps balls to ellipsoids with controlled eccentricity. Under certain mild integrability conditions, mappings of finite distortion are continuous, open and discrete, as are all quasiregular mappings by the Reshetnyak theorem. For continuous, open and discrete mappings between Euclidean n-domains the branch set, i.e. the set of points where the mapping fails to be a local homeomorphism, has topological dimension of at most $n-2$ by the Cernavskii-Vaisala theorem. For quasiregular mappings more properties for the branch set are known, but several important questions remain open. In this talk we show that an entire mappings of finite distortion cannot have a compact branch set when its distortion is locally finite and satisfies a certain asymptotic growth condition; $K(x) < o(\log (|x|))$. In particular this implies that the branch set of entire quasiregular mappings is either non-compact or empty. We furthermore show that the growth bound is asymptotically strict by constructing a continuous, open and discrete mapping of finite distortion from the Euclidean $n$-space to itself which is piecewise smooth, has a branch set homeomorphic to the$(n-2)$ torus and distortion arbitrarily close to the asymptotic bound $\log (|x|)$. The talk is based on joint work with Aapo Kauranen and Ville Tengvall.

Puzzles, flag manifolds, and Gromov-Witten invariants

Anders Skovsted Buch (Rutgers University)

Abstract: The development of algebraic geometry has been motivated by enumerative geometric questions where one asks for the number of geometric figures of some type that satisfy a list of conditions. For example, the Gromov-Witten invariants of a flag manifold counts the number of curves that meet a list of Schubert varieties in general position. Many enumerative problems can be reduced to understanding the Schubert structure constants of flag manifolds. Standard conjectures about #P-functions indicate that these structure constants are best expressed as the number of objects in some combinatorially defined set. The classical Littlewood-Richardson rule for the structure constants of Grassmannians is an example of this, but it is not known if all Schubert structure constants can be (reasonably) expressed in this way. I will speak about recent results that express Schubert structure constants as the number of puzzles that can be created from a given list of puzzle pieces, as well as relations to the Gromov-Witten invariants of Grassmannians.